In population dynamics a frequently encountered model for fish populations is based on an empirical equation called the Ricker equation (see Greenwell, 1984): $$ N_{n+1}=\alpha N_{n} e^{-\beta N_{0}} \text {, } $$ In this equation, \(\alpha\) represents the maximal growth rate of the organism and \(\beta\) is the inhibition of growth caused by overpopulation. (a) Show that this equation has a steady state $$ \bar{N}=\frac{\ln \alpha}{\beta} . $$ (b) Show that the steady state in (a) is stable provided that $$ |1-\ln \alpha|<1 \text {. } $$

In population dynamics, a steady state is when the population size remains constant over time. This means that the number of individuals at a given time, say day or year, is the same as at the next time period.
In mathematical terms, for the Ricker equation, the steady state condition is reached when \(N_{n+1} = N_n\) which we denote as \bar{N}.
We are given the Ricker equation: \ \ N_{n+1} = \alpha N_{n} e^{-\beta N_{n}}\. Plugging \bar{N} in place of \(N_n\), we get: \ \ \bar{N} = \alpha \bar{N} e^{-\beta \bar{N}}\
To simplify, divide both sides by \(\bar{N}\) yielding: \ \ 1 = \alpha e^{-\beta \bar{N}}\
Take the natural logarithm of both sides to solve for \(\bar{N}\):
We know \ \ln(1) = 0\, therefore: \ 0 = \ln(\alpha) - \beta \bar{N}\
Rearranging this equation gives us the steady state population as: \ \ \bar{N} = \frac{\ln \alpha}{\beta}\.
This is when the population does not change and is in a stable condition.

When we talk about stability in the context of a steady state, we mean whether small deviations from the steady state will die out over time, bringing the population back to the steady state.
For the Ricker equation's steady state found as \(\bar{N} = \frac{\ln \alpha}{\beta}\), we need to check the stability using the derivative condition.
Specifically, the steady state is stable if \(\left|\frac{dN_{n+1}}{dN_n} \right|_{N_n = \bar{N}} < 1\).
For our Ricker equation this derivative is given by: \ \ \frac{dN_{n+1}}{dN_n} = \alpha e^{-\beta N_n} (1 - \beta N_n)\.
Substituting \(\bar{N} = \frac{\ln \alpha}{\beta}\) into the derivative, we get: \ \frac{dN_{n+1}}{dN_n} \Big|_{N_n = \bar{N}} = \alpha e^{-\beta (\frac{\ln \alpha}{\beta})} (1 - \beta \frac{\ln \alpha}{\beta})\
Simplifying we get: \ \ \alpha e^{-\ln \alpha} (1 - \ln \alpha) = \alpha \frac{1}{\alpha} (1 - \ln \alpha)\ which simplifies to: \ \ \ \frac{dN_{n+1}}{dN_n} \Big|_{N_n = \bar{N}} = 1 - \ln \alpha\
The condition of stability is that the absolute value has to be less than 1: \ \ \left|1 - \ln \alpha \right| < 1\. This tells us that as long as this condition is satisfied, the population will return to the steady state even after small disturbances.

A natural logarithm, symbolized as \(\ln\), is a logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828.
Natural logarithms are widely used in growth models and equations, like the Ricker equation, because they have properties that simplify the mathematics involved.
For instance, in the step of solving for \(\bar{N}\) in the Ricker equation, taking the logarithm helped transform the exponential equation into a manageable linear form.
Here's a quick summary of important properties of natural logarithms:

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